Question

how to write a quadratic function with a graph that has x intercepts (2,0) and (-6,0) and a maximum point of (-2,-4)

Answer 1

have you covered parabolas yet?

Answer 2

in general, a quadratic equation is in the form

ax^2 + bx + c = 0.

So from the conditions given, you have to determine the value of a, b and c.
1. the function has x-intercepts at (4,0)and(-1,0), so we can simply write the function as f(x)=a(x-4)(x+1). also, it is mentioned that the function has an opening upward, so a is required to be a positive value

2.in the same way of number 1, we can write this function as f(x)=a(x-2)(x-6). notice that (4,12) is on the graph of the function, and therefore f(x) satisfies that a(4-2)(4-6)=12, so a=-3. consequently, the function is f(x)=-3(x-2)(x-6)=-3x^2+24x-36

1.opening upward, means a>0. It crosses two points (4,0) and (-1,0), so we can write y = a(x-4)(x+1)




2. y = a(x-2)(x-6) because it passes the points (2,0) and (6,0). Since it has maximum, then a<0 and since the maximum happens at (4,12), we can determine a by solving a(4-2)(4-6) = 12.

3. The quadratic graph only touch the x-intercept at (-3,0), so it has the form y = a(x+3)^2. Then it also passes through the point (0,18), we can determine a by solving 18 = a(0+3)^2

another solution for 3: why not set the function as f(x)=ax^2+bx+18 because of the y-intercept. now we have to imagine the that (-3,0) should be the lowest point of the graph of the function. so -b/2a=-3, (4a*18-b^2)/4a=0. a little bit more difficult, but ok as well. a=2, b=12. f(x)=2x^2+12x+18